Quantitative propagation of chaos for mean field Markov decision process with common noise

Abstract

We investigate propagation of chaos for mean field Markov Decision Process with common noise (CMKV-MDP), and when the optimization is performed over randomized open-loop controls on infinite horizon. We first state a rate of convergence of order M γ N , where M N is the mean rate of convergence in Wasserstein distance of the empirical measure, and γ P p0, 1s is an explicit constant, in the limit of the value functions of N -agent control problem with asymmetric open-loop controls, towards the value function of CMKV-MDP. Furthermore, we show how to explicitly construct pǫ `OpM γ N qq-optimal policies for the N -agent model from ǫ-optimal policies for the CMKV-MDP. Our approach relies on sharp comparison between the Bellman operators in the N -agent problem and the CMKV-MDP, and fine coupling of empirical measures.